Banach Journal of Mathematical Analysis

On a reverse of Ando--Hiai inequality

Yuki Seo

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In this paper, we show a complement of Ando--Hiai inequality: Let $A$ and $B$ be positive invertible operators on a Hilbert space $H$ and $\alpha\in [0,1]$. If $A\ \sharp_{\alpha}\ B \leq I$, then $$A^r \, \sharp_\alpha \, B^r \leq \|(A \,\sharp_\alpha \, B)^{-1}\|^{1-r} I \quad \text{for all }\, 0 <r \leq 1, $$ where $I$ is the identity operator and the symbol $\| \cdot \|$ stands for the operator norm.

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Banach J. Math. Anal., Volume 4, Number 1 (2010), 87-91.

First available in Project Euclid: 27 April 2010

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Zentralblatt MATH identifier

Primary: 47A63: Operator inequalities
Secondary: 47A30: Norms (inequalities, more than one norm, etc.) 47A64: Operator means, shorted operators, etc.

Ando--Hiai inequality positive operator geometric mean


Seo, Yuki. On a reverse of Ando--Hiai inequality. Banach J. Math. Anal. 4 (2010), no. 1, 87--91. doi:10.15352/bjma/1272374672.

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